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Definition: Soundness and Completeness of a Logical Calculus

A logical calculus $L$ is called

• sound, if any derivable statement is also a valid statement, formally $\vdash \phi$ implies $\models \phi,$
• complete, if any valid statement is also derivable, formally $\models \phi$ implies $\vdash \phi.$

Notes:

• Soundness and completeness are semantical properties of a logical calculus.
• A logical calculus is sound if all its theorems are true statements, and it is complete if all true statements are theorems.
• Soundness and completeness are desirable properties of logical calculi since such systems induce no difference between the provability and the truth of theorems.

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References

Bibliography

1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
2. Beierle, C.; Kern-Isberner, G.: "Methoden wissensbasierter Systeme", Vieweg, 2000